Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Verification of Entrance Dose Measurements with Thermoluminescent Dosimeters in Conventional Radiotherapy Procedures Delivered with Co‑60 Teletherapy Machine

Background: The use of in vivo dosimetry with thermolumiscent dosimeters (TLDs) as a veritable means of quality control in conventional radiotherapy procedures was determined in this work. Aim: The objective of this study was to determine the role of in vivo dosimetry with thermoluminescent dosimeters (TLDs) as part of quality control and audit in conventional radiotherapy procedures delivered with Co‑60 teletherapy machine. Subjects and Methods: Fifty‑seven patients with cancers of the breast, pelvis, head and neck were admitted for this study. TLD system at the Radiation Monitoring and Protection Centre, Lagos State University, Ojo, Lagos‑Nigeria was used for the in vivo entrance dose readings. All patients were treated with Co‑60 (T780c) teletherapy machine at 80 cm source to surface distance located at Eko Hospitals, Lagos. Two TLDs were placed on the patient surface within 1 cm from the center of the field of treatment. Build‑up material made of paraffin wax with a density of 0.939 g/cm3 and a thickness 0.5 cm was placed on top of the TLDs. A RADOS RE 200 TLD reader was used to read out the TLDs over 12 s and at a temperature of 300°C. Results: The results showed that there was no significant difference between the expected dose and measured dose of breast (P = 0.11), H and N (P = 0.52), and pelvis (P = 0.31) patients. Furthermore, percentage difference between expected dose and measured dose of the three treatment sites were not significantly different (P = 0.11). More so, 88.9% (16/18) treated breast, 91.3% (21/23) pelvis, and 86.7% (13/15) H and N patients had percentage deviation difference less than 5%. In general, 89.3% (50/56) patients admitted for this study had their percentage deviation difference below 5% recommended standard limit. Conclusion: The values obtained establish that there are no major differences from similar studies reported in literature. This study was also part of quality control and audit of the radiotherapy procedures in the center as expected by national and international regulatory bodies.Keywords: Co‑60 machine in vivo dosimetry, Conventional radiotherapy, Entrance dose, Thermoluminescent dosimeter

Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Rehabilitating mangrove ecosystem services: a case study on the relative benefits of abandoned pond reversion from Panay Island, Philippines

Mangroves provide vital climate change mitigation and adaptation (CCMA) ecosystem services (ES), yet have suffered extensive tropics-wide declines. To mitigate losses, rehabilitation is high on the conservation agenda. However, the relative functionality and ES delivery of rehabilitated mangroves in different intertidal locations is rarely assessed. In a case study from Panay Island, Philippines, using field- and satellite-derived methods, we assess carbon stocks and coastal protection potential of rehabilitated low-intertidal seafront and mid- to upper-intertidal abandoned (leased) fishpond areas, against reference natural mangroves. Due to large sizes and appropriate site conditions, targeted abandoned fishpond reversion to former mangrove was found to be favourable for enhancing CCMA in the coastal zone. In a municipality-specific case study, 96.7% of abandoned fishponds with high potential for effective greenbelt rehabilitation had favourable tenure status for reversion. These findings have implications for coastal zone management in Asia in the face of climate change

A general lower bound for collaborative tree exploration

We consider collaborative graph exploration with a set of $k$ agents. All agents start at a common vertex of an initially unknown graph and need to collectively visit all other vertices. We assume agents are deterministic, vertices are distinguishable, moves are simultaneous, and we allow agents to communicate globally. For this setting, we give the first non-trivial lower bounds that bridge the gap between small ($k \leq \sqrt n$) and large ($k \geq n$) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains. First, we significantly extend a lower bound of $\Omega(\log k / \log\log k)$ by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range $k \leq n \log^c n$ for any $c \in \mathbb{N}$. Second, we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm. In particular, we show that any collaborative tree exploration algorithm with $k = Dn^{1+o(1)}$ agents has a competitive ratio of $\omega(1)$, while Dereniowski et al. gave an algorithm with $k = Dn^{1+\varepsilon}$ agents and competitive ratio $O(1)$, for any $\varepsilon > 0$ and with $D$ denoting the diameter of the graph. Lastly, we show that, for any exploration algorithm using $k = n$ agents, there exist trees of arbitrarily large height $D$ that require $\Omega(D^2)$ rounds, and we provide a simple algorithm that matches this bound for all trees

Psychological type and prayer preferences: a study among Anglican clergy in the United Kingdom

This study applies the framework of Jungian psychological type theory to define eight aspects of prayer preference, namely: introverted prayer, extraverted prayer, sensing prayer, intuitive prayer, feeling prayer, thinking prayer, judging prayer, and perceiving prayer. On the basis of data provided by 1,476 newly ordained Anglican clergy from England, Ireland, Scotland, and Wales, eight 7-item scales were developed to access these aspects of prayer preferences. Significant correlations were found between each prayer preference and the relevant aspect of psychological type accessed by the Keirsey Temperament Sorter. These data support the theory that psychological type influences the way in which people pray

Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

Estimation of the Optimal Statistical Quality Control Sampling Time Intervals Using a Residual Risk Measure

Background: An open problem in clinical chemistry is the estimation of the optimal sampling time intervals for the application of statistical quality control (QC) procedures that are based on the measurement of control materials. This is a probabilistic risk assessment problem that requires reliability analysis of the analytical system, and the estimation of the risk caused by the measurement error. Methodology/Principal Findings: Assuming that the states of the analytical system are the reliability state, the maintenance state, the critical-failure modes and their combinations, we can define risk functions based on the mean time of the states, their measurement error and the medically acceptable measurement error. Consequently, a residual risk measure rr can be defined for each sampling time interval. The rr depends on the state probability vectors of the analytical system, the state transition probability matrices before and after each application of the QC procedure and the state mean time matrices. As optimal sampling time intervals can be defined those minimizing a QC related cost measure while the rr is acceptable. I developed an algorithm that estimates the rr for any QC sampling time interval of a QC procedure applied to analytical systems with an arbitrary number of critical-failure modes, assuming any failure time and measurement error probability density function for each mode. Furthermore, given the acceptable rr, it can estimate the optimal QC sampling time intervals

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels